quantitative analysis on low cycle fatigue damage: a microstructural model for the prediction of...

Materials Science and Engineering A 379 (2004) 210–217

Quantitative analysis on low cycle fatigue damage: a microstructuralmodel for the prediction of fatigue life

H.J. Kima, C.S. Leea,∗, S.H. Parkb, D.H. Shinc

a Department of Materials Science and Engineering, Pohang University of Science and Technology, Pohang 790-784, Republic of Koreab POSCO Automotive Steels Application Center, Gwangyang 545-711, Republic of Korea

c Department of Metallurgy and Materials Science, Hanyang University, Ansan, Kyunggi-Do 425-791, Republic of Korea

Received 31 October 2003; received in revised form 21 January 2004

Abstract

A study has been made to develop a model predicting the low cycle fatigue life of a material in relation to its microstructural variables. Toachieve this goal, the concept of damage accumulation by multiple surface cracks has been adopted. An equation for stage I crack growthsuggested by Tomkins was modified to consider the effect of grain size on the crack growth rate at early stage, and statistical analysis was carriedout to calculate the final crack length for fatal failure. A concept of equivalent crack length has been used to present the quantitative descriptionof crack growth rate when multiple cracks grow at the same time. To verify the suggested model, low cycle fatigue tests were conducted forthe polycrystalline single-phase steel with the various grain sizes. The results showed a good agreement between the experimental data andthe predicted curve.© 2004 Elsevier B.V. All rights reserved.

Keywords: LCF life prediction; Multi-cracking; Equivalent crack; Crack distribution; Grain size

1. Introduction

To prevent failures of materials loaded cyclically witha high plastic strain amplitude, low cycle fatigue (LCF)was widely investigated not only in the experimental works[1–4] but also in the development of accurate life predictionmodels[5–7]. During the last several decades, many modelsfor predicting LCF life were proposed using macroscopicparameters such as elastic modulus[8], strain density[9],plastic strain energy[10] and total strain energy[11]. How-ever, the earlier works did not consider the effect of mi-crostructural factors such as grain size and volume fractionof second phase particles in their models due to the com-plexity of microstructures of various materials. Consideringthe fact that the mechanical properties including low cyclefatigue resistance are greatly influenced by the microstruc-ture[4,12–14], there is a strong need to reflect the effects ofmicrostructural parameters in the LCF life prediction model.

∗ Corresponding author. Tel.:+82-54-279-2141;fax: +82-54-279-2399.

E-mail address: [email protected] (C.S. Lee).

Accordingly, efforts were focused to establish the relation-ships between microstructure and damage of fatigue process.

The fatigue crack growth rate (da/dN) is generally con-sidered the most common parameter relating the damageof fatigue process with the microstructure because a crackgrows by dislocation movement which is largely influencedby the microstructural features[12–14]. Since Shanley[15]introduced a damage theory by defining the crack length asa damage measure, many quantitative analyses were madebased on the concepts derived from dislocation theory anda synthesis of the macroscopic elasto-plastic fracture theory[16–19].

Recently, a multi-cracking mechanism is receiving a greatinterest in explaining the detailed crack growth behaviorduring the LCF process[17–28]. When a material receivescyclic loads with high plastic strain, many small cracks nu-cleate and grow simultaneously interacting to each other[22–28]. Finally a dominant crack forms from such interac-tions and leads to final fracture[22–28]. This multi-crackingmechanism is also considered important in the prediction ofLCF life. However, there are many difficulties in describingthe accurate fatigue crack growth rate due to the simultane-ous growth and interaction of small cracks.

0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.msea.2004.01.044

H.J. Kim et al. / Materials Science and Engineering A 379 (2004) 210–217 211

Fig. 1. Optical micrographs showing the initial three microstructures of the tested steel: (a) as received (6.9�m), (b) 750◦C, 1 h (10.3�m) and (c)900◦C, 2 h (17.1�m).

Polák et al.[25] reported characteristic features of thegrowth behavior of short natural cracks in 316L steel andaluminium–lithium alloys. A high density of fatigue microc-racks was generated in 316L steel, but in most experimentalconditions only one of them became a dominant crack lead-ing to fracture. To analyze the crack growth rate in the caseof multi-cracking, Polák[25] introduced the concept of ‘anequivalent crack,’ which always indicated the longest crackpropagating under conditions of external loading. Conse-quently, the crack growth rate of the longest crack during thelow cycle fatigue process was successfully described, but themicrostructural influences on the growth rate of small crackand the fatal crack length was not considered in their work.

Therefore, in this study, a LCF life prediction model re-flecting the influence of grain size was proposed by modify-ing the Polák’s crack growth equation. To calculate the fatalcrack length, statistical analysis was applied to determine thedistribution of multiple cracks in relation with the grain size.

2. Experimental details

Along with an effort to develop a theoretical model forpredicting low cycle fatigue life, experiments of low cy-

cle fatigue test were also carried out to validate the pro-posed model. Specimens used in this study were polycrys-talline single-phase steel with three different grain sizes.The chemical composition of the steel is 0.17C, 0.94 Mnin weight percent and balanced Fe. To vary the grain size,the as-received microstructure with an initial grain size of6.9�m was heat-treated at 750◦C for 1 h and 900◦C for2 h, respectively, followed by air cooling. The resulting mi-crostructures are shown inFig. 1. The average grain sizesand tensile properties are listed inTable 1. Sheet type speci-mens were prepared for the tensile and LCF tests. The gaugedimension of the specimens was 6 mm×6 mm×3 mm. Be-fore running an actual LCF test, the surface of gauge re-gion was mechanically polished using abrasive papers up to#2000 to give a smooth surface. All the tests were carried

Table 1The room temperature tensile properties of the tested steels

Averagegrain size(�m)

Yieldstrength(MPa)

Ultimatestrength(MPa)

Elongation(%)

As received 6.9 353 485 47.3750◦C for 1 h 10.3 297 433 56.5900◦C for 2 h 17.1 231 430 52.8

212 H.J. Kim et al. / Materials Science and Engineering A 379 (2004) 210–217

out at room temperature in air using a computer controlledservo hydraulic test machine. The LCF test was performedwith the applied total strain amplitude (�εt) ranging from3× 10−3 to 7× 10−3 at a constant strain rate of 2× 10−3 sand the strain ratio of−1. Triangular waveform was used inthis study. To measure the crack lengths of individual cracks,cellulose acetate replicas were attached on the gauge regionof a specimen before cycling. The test was sequentially in-terrupted after several hundreds of cycles, and the lengthsof every cracks shown in the replicas were measured. Byrepeating the steps, it was possible to measure the crackgrowth rate of the longest crack.

3. Proposed model to predict LCF life

Generally, the LCF life is divided into the nucleation ofmulticracks, the growth and coalescence of multicracks andthe growth of a fatal crack to final failure[22–25,28]. It wasreported that over 80% of the entire LCF life (Nf ) was spentin the growth and coalescence of multicracks before the for-mation of a fatal crack[22–25,28]. The stage of growth andcoalescence of multicracks was reported as stage I regimewhere the crack growth was performed by mode II shearcracking[25]. The fatigue process was changed from stage Ito stage II as the length of small crack reached some criticalvalue. In the stage II, the crack growth mode was changedfrom mode II shear cracking to mode I normal cracking. Thelife time spent during the stage II could be analyzed usingthe well-known Paris law[29], which was not significant ascompared to that of entire life[25]. Therefore, it could beassumed that the total LCF life was mostly spent in the stage

Fig. 2. LCF life with the variation of grain size.

I crack growth regime. Using the above assumption, the lifespent for the stage I crack growth regime was calculated toapproximate the total LCF life in this study.

The crack growth rate was assumed to depend on plasticstrain range. Basinski et al.[26] observed a constant growthrate for crystallographic growth of a small fatigue crackand correlated experimental crack growth rate with the ap-plied plastic strain amplitude using an equation proposed byTomkins[27] as shown inEq. (1).

Stage I :da

dN= A(�εp)

B (1)

where�εp is the applied plastic strain amplitude andA andB are the material constants. However, there is a difficultyin applying theEq. (1)directly to calculate the crack growthrate under LCF condition because multiple cracks initiateand grow at the same time. As mentioned in the earlier, theconcept of ‘equivalent crack’ was utilized. Applying thisconcept toEq. (1), the growth and the coalescence of mul-tiple cracks could be presented as

daeq

dN= A(�εp)

B (2)

whereaeq is the length of equivalent crack representing thelongest one at each fatigue cycle.

To consider the effect of grain size, a modification forthe above equation was made in the following manner. It iswell-known that plastic strain at the crack tip acts as a driv-ing force for the propagation of small crack[25]. For a givenamount of plastic strain amplitude, the resolved shear strainon a specific slip system of each grain will be different de-pending on the grain size as well as the grain orientation.


Since the slip band is typically blocked by the grain bound-ary for the small crack[24], the amount of plastic strain atthe crack tip will be larger in the large grain size material. Asa result, the effective driving force for the crack propagationwill be larger for the specimen with the larger grain size.However, it may not be linearly proportional to the grainsize because of suppresion by the neighboring grains. Stroh[29] has proposed that the shear stress acting on the grainboundary by pile-up dislocations linearly increases with thevalue of

√d. Considering a lower limit in the grain size

(d0) where the theory of dislocation pile-up is not valid, themodified equation of stage I crack growth can be expressedas follows.

Fig. 3. Surface crack propagation curves of the tested steel showing the effect of (a) grain size and (b) total strain amplitude.

daeq

dN= C

(√d

d0�εp

)m

= C′(√

d�εp)m (3)

whereC′ is the material constant.IntegratingEq. (3) from aeq = 0 to aeq = af will results

in the total life.

Nf =∫ af

0

daeq

C′(√

d�εp)m(4)

where af is the crack length for the formation of a fatalcrack. In order to obtain the value ofaf , statistical analy-sis was performed considering the distribution of multiple


short cracks. It is assumed that the nucleation of an individ-ual crack is a random uncorrelated process, which impliesthat there is no interaction between the cracks at the nu-cleation stage. For a given material and an imposed cyclicstrain amplitude, there exists a specific value of probability(P) that a grain with the size ofd contains a transgranularcrack at the grain interior for each cycle. For a specific area(S) of specimen surface, the average number of grains areS/d2. Therefore, the number of grains that have a trangranu-lar crack on the area (S) of interest arePS /d2. Based on thepercolation theory, Brechet et al.[30] proposed the proba-bility that a two-dimensional system can have a crack witha size ofaeq/d (in units of the grain size,aeq ≥ d) as

P(aeq) = Paeq/d(1 − P)2aeq/d (5)

In the above equation, Brechet et al. considered all thesurrounding grains of a grain containing a crack. When it isassumed in this study that the linkage of cracks occurs alongthe direction perpendicular to the loading axis, the exponentof 2aeq/d in the above equation can be simplified as 2. Then,Eq. (5)can be changed to

P(aeq) = Paeq/d(1 − P)2 (6)

Eq. (6) indicates the probability for existence of smallcrack with the size ofaeq/d. The total number of such cracksis proportional to the number of grains in the surfaceS.

S

d2P(aeq) = S

d2Paeq/d(1 − P)2 (7)

A fatal crack is formed immediately after cracks start tointeract with each other and grow very fast leading to failure.Earlier works[22–26] have reported that there is only onefatal crack in the whole system containingS/d2 grains. Thevalue ofEq. (7)can be set as 1 when a fatal crack with thelength of af is formed. Then, the condition of fatal crackformation is given by

S

d2Paf /d(1 − P)2 = 1 (af : the length of a fatal crack)(8)

FromEq. (8),

af = dln(d2/S)

ln(P)(9)

The probability (P) is proportional to the number of crackinitiation sites per unit area, which is linearly proportionalto �εp [30]. Therefore,Eq. (9)can be written as

af = dln(d2/S)

ln(C′′�εp)(10)

where C′′ is the material constant. Considering the effectof grain size,�εp should be replaced by

√d�εp. From

the Eqs. (4) and (10), the total life can be obtained by theEq. (11).

Nf = d ln(d2/S)

C(√

d�εp)m ln(C′√d�εp)(11)

Table 2The transition crack length of the specimens with different grain size

Grain size (�m) Transition cracklength (�m)

Transition cracklength/grain size

6.9 100 14.510.3 178.9 17.417.1 336.8 19.7

Table 3The iterated constants inEq. (11) for the steels with the grain sizes of6.9 and 17.1�m

Grain size (�m) (iterated) C C′ m

6.9 9300 5.7× 10−5 3.817.1 9300 3.49× 10−3 3.8

4. Verification of the model with the experimentalresults

Three microstructures with different grain sizes are shownin Fig. 1 and the tensile properties are listed inTable 1.Fig. 2 shows the plot of total strain amplitude (�εt) versusreversals to failure (2Nf ) with the variation of grain size. TheLCF life was found to increase with the decrease of grainsize. This behavior agreed well with experimental resultsobtained by other work[28]. The shorter life of a coarsegrained material was mainly attributed to the reduction oflife at an early stage of crack growth[28]. To clarify theeffect of grain size on the early crack growth behavior, smallcrack growth test was conducted.Fig. 3 shows the plot ofequivalent crack length versus the normalized fatigue life(N/Nf ) with the variation of grain size (Fig. 3(a)) and appliedtotal strain amplitude (Fig. 3(b)). It is apparent inFig. 3(a)that more than 70–80% of total LCF life has been spent inthe stage I crack growth regime representing the region oflinear increase in crack length. It is also shown inFig. 3thatthe transition crack length (at, i.e. the crack length showingrapid increase in the crack growth rate) increases with theincrease of grain size and the effect of total strain amplitudeon at is not significant. The average value ofat was foundto be 10–20 times of the grain size (Table 2).

Several unknown constants (C, C′, m) in Eq. (11)weredetermined by iteration method using the experimental LCFdata for two microstructures of 6.9 and 17.1�m grain sizes(Table 3). For the fundamental study, the physical meaningand the relation with microstructural variable (grain size inthis study) should be clarified for each constant. The con-stantsC andm were derived from the crack growth rate andthe constantC′ was introduced by the statistical distributionof small cracks. From the result of iteration, the constantsC

Table 4The predicted constants inEq. (11)for the steel with grain size of 10.3�m

Grain size (�m) (predicted) C C′ m

10.3 9300 3.51× 10−4 3.8


andm were found to have fixed values with the variation ofgrain size. But the constantC′ was found to vary with thevariation of grain size. In order to analyze the value ofC′with the grain size, a relationship between the grain size andthe probability distribution parameter of small cracks shouldbe obtained. To estimate the scatter characteristics in a quan-titative manner, the distribution properties for each fatiguedata should be analyzed. The distribution functionF(x) de-fined as the probability of a random variable was adopted toexpress the distribution properties for the variable. Previousstudies concerning the statistical treatment of fatigue life andcrack length suggested that a Weibull distribution[32,33]ora log-normal distribution[34] was suitable for the analysis

Fig. 4. Plot of total strain amplitude (�εt) vs. LCF life for the tested steels with the grain size of (a) 6.9�m and (b) 17.1�m.

of fatigue data[32–34]. In this study, a Weibull distributionfunction was applied for the statistical analysis of small crackdistribution with respect to the grain size, and was shown asfollowing.

F(x) = 1 − exp

[−(

x − γ

η

)m](12)

Here, three constants (m, η, γ) are the shape, scale and lo-cation parameters, respectively. In this study, the variablexwas replaced by the average grain sized. Assuming that ev-ery grain had a transgranular crack at the interior of a grain,the location parameterγ indicating the minimum value ofgrain size that could have the cyclic damage accumulation


Fig. 5. Plot of total strain amplitude (�εt) vs. LCF life for the tested steel with the grain size of 10.3�m.

could be set to zero. Other two parameters,m andη couldbe obtained by the iteration method to evaluate the LCF life.Therefore, the constantC′ could be obtained asEq. (13).

C′ = 1 − exp[−(0.0168× d)4.6353] (13)

Fig. 4 shows the experimental and iteration results forfatigue lives of two different microstructures with grainsizes of 6.9 and 17.1�m. Two unknown constantsC andm were obtained by iteration and presented inTable 3. Thevalue ofm was generally known to be 2–4 for low carbonsteels[31]. Our result (m = 3.8) were well accorded withother investigation. Using theEq. (13), the values ofC′were obtained for various grain size of 10.3�m (Table 4).Using the data ofTable 4, the LCF life was predicted forthe grain size 10.3�m, and compared with the experi-mental results. As shown inFig. 5, a good agreement wasobserved between the experimental data and the predictedcurve.

5. Summary

A microstructural model for predicting the low cycle fa-tigue life was developed by modifying an equation for stageI crack growth (suggested by Tomkins) and by consideringdamage accumulation from multiple cracking. In order tocalculate the length of a fatal crack, statistical analysis wasperformed considering the distribution of multiple shortcracks and a Weibull distribution function. Experimental lowcycle fatigue tests for steels with three different grain sizeswere carried out to verify the suggested model. The pre-dicted curve was in good accordance with the experimentaldata.

Acknowledgements

This research was partly supported by a grant through2003 National Research Laboratory program funded by theMinistry of Science and Technology, Korea, and also partlysupported by a grant from the Center for Advanced Materi-als Processing (CAMP) of the 21st Century Frontier R&DProgram funded by the Ministry of Science and Technology,Korea.

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quantitative analysis on low cycle fatigue damage: a microstructural model for the prediction of...

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